 Solution

Use sinusoidal phasors, $v(t)=\mathrm{Re}(\tilde{V}{e}^{i\omega t})$. Differentiation becomes multiplication by $i\omega$, so the time-dependent element laws reduce to algebraic complex impedances:Then $\tilde{V}=Z\tilde{I}$ encodes both amplitude ratio $|Z|$ and voltage-current phase difference $\arg Z$. Series impedances add, while parallel admittances add. This applies to linear sinusoidal steady state; switching transients are separate homogeneous solutions.